The research into curvilinear flight synthetic aperture radar (CF-SAR) is the inevitable result of the comprehensive practicality of SAR. The flight path of the SAR platform in real applications, which is highly nonlinear or curvy due to three-dimensional velocity and acceleration, cannot be described by the traditional uniform linear motion model. New mathematical models, signal characteristics, imaging algorithms, and system design criteria must be proposed and investigated for CF-SAR. This paper provides a comprehensive overview of CF-SAR. Firstly, the basic concept, unified model, and general signal characteristics of CF-SAR are defined, derived, and analyzed, respectively. Additionally, the advantages and drawbacks of current methodologies are reviewed. Discussions on the CF-SAR’s applications are presented from the perspective of typical platforms, new configurations, and advanced technologies, which are suitable means to fulfill the increasing user requirements. Finally, the challenges faced by CF-SAR are summarized, and some future trends for the study of CF-SAR are explored. Hopefully, this paper will serve as a reference for SAR researchers/engineers and stimulate the future development and actual application of CF-SAR.